Solo dwarfs - III. Exploring the orbital origins of isolated Local Group galaxies with Gaia Data Release 2

We measure systemic proper motions for distant dwarf galaxies in the Local Group and investigate if these isolated galaxies have ever had an interaction with the Milky Way or M31. We cross-match photometry of isolated, star-forming, dwarf galaxies in the Local Group, taken as part of the Solo survey, with astrometric measurements from Gaia Data Release 2. We find that NGC 6822, Leo A, IC 1613, and WLM have sufficient supergiants with reliable astrometry to derive proper motions. An additional three galaxies (Leo T, Eridanus 2, and Phoenix) are close enough that their proper motions have already been derived using red giant branch stars. Systematic errors in Gaia DR2 are significant for NGC 6822, IC 1613, and WLM. We explore the orbits for these galaxies, and conclude that Phoenix, Leo A, and WLM are unlikely to have interacted with the Milky Way or M31, unless these large galaxies are very massive ( ${\gtrsim}1.6 \times 10^{12}\, \mathrm{M}_\odot$ ). We rule out a past interaction of NGC 6822 with M31 at ${\sim}99.99{{\ \rm per\ cent}}$ confidence, and find there is a <10 per cent chance that NGC 6822 has had an interaction with the Milky Way. We examine the likely origins of NGC 6822 in the periphery of the young Local Group, and note that a future interaction of NGC 6822 with the Milky Way or M31 in the next 4 Gyr is essentially ruled out. Our measurements indicate that future Gaia data releases will provide good constraints on the interaction history for the majority of these galaxies.